Let’s assume that is continuous and positive on the interval . Then the definite integral represents the area of the region bounded by the graph of and the x-axis, from x = a to x = b. First, we partition the interval into n subintervals, each of width such that Then we can form a trapezoid for each subinterval and the area of the ith trapezoid = . This implies that the sum of the areas of the n trapezoids is Area =

# Month: August 2013

# Exercise on Relations

Let S and S‘ be the following subsets of the plane: and

**a)** Show that S’ is an equivalence relation on the real line and that .

**Proof: Reflexivity**–

** Symmetry**–

** Transitivity- **If then and thus z-y is the difference of two integers which implies that z-y is itself an integer.

To show that we note that which

**b) **Show that given any collection of equivalence relations on a set A, their intersection is an equivalence relation in A.

**Proof: **Let be a nonempty class of equivalence relations and let

**Reflexivity- **If then .

**Symmetry**– .

**Transitivity**– If then the intersection is an equivalence relation on A.